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Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem

Thomas Wick ORCID logo

Abstract

In this work, goal-oriented adjoint-based a posteriori error estimates are derived for a nonlinear phase-field discontinuity problem in which a scalar-valued displacement field interacts with a scalar-valued smoothed indicator function. The latter is subject to an irreversibility constraint, which is regularized using a simple penalization strategy. The main advancements in the current work are error identities, resulting estimators, and two-sided estimates employing the dual-weighted residual method, which address the influence of the phase-field regularization, penalization, and spatial discretization parameters. Some numerical tests accompany our derived estimates.

MSC 2010: 74R10; 65N30; 65N50; 49M15; 35Q74

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: 392587580

Funding statement: The author is supported by the German Research Foundation, Priority Program 1748 (DFG SPP 1748) named Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis in the sub-project (WI 4367/2-1) under the project id 392587580. The author was partially supported by the Sino-German Center for Research Promotion under Grant GZ 1571.

References

[1] M. Ainsworth, J. T. Oden and C.-Y. Lee, Local a posteriori error estimators for variational inequalities, Numer. Methods Partial Differential Equations 9 (1993), no. 1, 23–33. 10.1002/num.1690090104Search in Google Scholar

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University, New York, 2000. Search in Google Scholar

[3] L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence, Comm. Pure Appl. Math. 43 (1990), no. 8, 999–1036. 10.1002/cpa.3160430805Search in Google Scholar

[4] L. Ambrosio and V. M. Tortorelli, On the approximation of free discontinuity problems, Boll. Unione Mat. Ital. B (7) 6 (1992), no. 1, 105–123. Search in Google Scholar

[5] J. Andersson and H. Mikayelyan, The asymptotics of the curvature of the free discontinuity set near the cracktip for the minimizers of the Mumford–Shah functional in the plain, preprint (2015), https://arxiv.org/abs/1204.5328v2. Search in Google Scholar

[6] D. Arndt, W. Bangerth, T. C. Clevenger, D. Davydov, M. Fehling, D. Garcia-Sanchez, G. Harper, T. Heister, L. Heltai, M. Kronbichler, R. M. Kynch, M. Maier, J.-P. Pelteret, B. Turcksin and D. Wells, The deal.II library. Version 9.1, J. Numer. Math. 27 (2019), no. 4, 203–213. 10.1515/jnma-2019-0064Search in Google Scholar

[7] D. Arndt, W. Bangerth, D. Davydov, T. Heister, L. Heltai, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin and D. Wells, The deal.II finite element library: Design, features, and insights, Comput. Math. Appl. 81 (2021), 407–422. 10.1016/j.camwa.2020.02.022Search in Google Scholar

[8] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2003. 10.1007/978-3-0348-7605-6Search in Google Scholar

[9] R. E. Bank, A. Parsania and S. Sauter, Saturation estimates for h p -finite element methods, Comput. Vis. Sci. 16 (2013), no. 5, 195–217. 10.1007/s00791-015-0234-2Search in Google Scholar

[10] R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: Basic analysis and examples, East-West J. Numer. Math. 4 (1996), no. 4, 237–264. Search in Google Scholar

[11] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer. 10 (2001), 1–102. 10.1017/S0962492901000010Search in Google Scholar

[12] A. Bonnet and G. David, Cracktip is a Global Mumford–Shah Minimizer, Astérisque 274, Société Mathématique de France, Paris, 2001. Search in Google Scholar

[13] B. Bourdin, G. A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture, J. Mech. Phys. Solids 48 (2000), no. 4, 797–826. 10.1016/S0022-5096(99)00028-9Search in Google Scholar

[14] B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture, J. Elasticity 91 (2008), no. 1–3, 5–148. 10.1007/978-1-4020-6395-4Search in Google Scholar

[15] M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, Multiscale Model. Simul. 1 (2003), no. 2, 221–238. 10.1137/S1540345902410482Search in Google Scholar

[16] F. Brezzi, W. W. Hager and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math. 28 (1977), no. 4, 431–443. 10.1007/BF01404345Search in Google Scholar

[17] F. Brezzi, W. W. Hager and P.-A. Raviart, Error estimates for the finite element solution of variational inequalities. II. Mixed methods, Numer. Math. 31 (1978/79), no. 1, 1–16. 10.1007/BF01396010Search in Google Scholar

[18] G. F. Carey and J. T. Oden, Finite Elements. Vol. III. Computational Aspects, Prentice Hall, Englewood Cliffs, 1984. Search in Google Scholar

[19] C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 (2014), no. 6, 1195–1253. 10.1016/j.camwa.2013.12.003Search in Google Scholar PubMed PubMed Central

[20] C. Carstensen, D. Gallistl and J. Gedicke, Justification of the saturation assumption, Numer. Math. 134 (2016), no. 1, 1–25. 10.1007/s00211-015-0769-7Search in Google Scholar

[21] C. Carstensen and J. Hu, An optimal adaptive finite element method for an obstacle problem, Comput. Methods Appl. Math. 15 (2015), no. 3, 259–277. 10.1515/cmam-2015-0017Search in Google Scholar

[22] C. Carstensen and R. Verfürth, Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal. 36 (1999), no. 5, 1571–1587. 10.1137/S003614299732334XSearch in Google Scholar

[23] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1987. Search in Google Scholar

[24] P. Deuflhard, Newton Methods for Nonlinear Problems, Springer Ser. Comput. Math. 35, Springer, Heidelberg, 2011. 10.1007/978-3-642-23899-4Search in Google Scholar

[25] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. 10.1137/0733054Search in Google Scholar

[26] W. Dörfler and R. H. Nochetto, Small data oscillation implies the saturation assumption, Numer. Math. 91 (2002), no. 1, 1–12. 10.1007/s002110100321Search in Google Scholar

[27] B. Endtmayer, Multi-goal oriented a posteriori error estimates for nonlinear partial differential equations, PhD thesis, Johannes Kepler University Linz, 2021. Search in Google Scholar

[28] B. Endtmayer, U. Langer and T. Wick, Multigoal-oriented error estimates for non-linear problems, J. Numer. Math. 27 (2019), no. 4, 215–236. 10.1515/jnma-2018-0038Search in Google Scholar

[29] B. Endtmayer, U. Langer and T. Wick, Two-side a posteriori error estimates for the dual-weighted residual method, SIAM J. Sci. Comput. 42 (2020), no. 1, A371–A394. 10.1137/18M1227275Search in Google Scholar

[30] B. Endtmayer, U. Langer and T. Wick, Reliability and efficiency of dwr-type a posteriori error estimates with smart sensitivity weight recovering, Comput. Methods Appl. Math. 21 (2021), no. 2, 351–371. 10.1515/cmam-2020-0036Search in Google Scholar

[31] R. S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput. 28 (1974), no. 128, 963–971. 10.1090/S0025-5718-1974-0391502-8Search in Google Scholar

[32] M. Feischl, D. Praetorius and K. G. van der Zee, An abstract analysis of optimal goal-oriented adaptivity, SIAM J. Numer. Anal. 54 (2016), no. 3, 1423–1448. 10.1137/15M1021982Search in Google Scholar

[33] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998), no. 8, 1319–1342. 10.1016/S0022-5096(98)00034-9Search in Google Scholar

[34] C. Grossmann and H.-G. Roos, Numerical Treatment of Partial Differential Equations, Universitext, Springer, Berlin, 2007. 10.1007/978-3-540-71584-9Search in Google Scholar

[35] T. Heister, M. F. Wheeler and T. Wick, A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach, Comput. Methods Appl. Mech. Engrg. 290 (2015), 466–495. 10.1016/j.cma.2015.03.009Search in Google Scholar

[36] T. Heister and T. Wick, Parallel solution, adaptivity, computational convergence, and open-source code of 2d and 3d pressurized phase-field fracture problems, PAMM. Proc. Appl. Math. Mech. 18 (2018), 10.1002/pamm.201800353. 10.1002/pamm.201800353Search in Google Scholar

[37] T. Heister and T. Wick, pfm-cracks: A parallel-adaptive framework for phase-field fracture propagation, Software Impacts 6 (2020), Article ID 100045. 10.1016/j.simpa.2020.100045Search in Google Scholar

[38] D. W. Kelly, J. P. D. S. R. Gago, O. C. Zienkiewicz and I. Babuška, A posteriori error analysis and adaptive processes in the finite element method. I. Error analysis, Internat. J. Numer. Methods Engrg. 19 (1983), no. 11, 1593–1619. 10.1002/nme.1620191103Search in Google Scholar

[39] N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Stud. Appl. Math. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1988. 10.1137/1.9781611970845Search in Google Scholar

[40] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics Appl. Math. 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000. 10.1137/1.9780898719451Search in Google Scholar

[41] R. Kornhuber, A posteriori error estimates for elliptic variational inequalities, Comput. Math. Appl. 31 (1996), no. 8, 49–60. 10.1016/0898-1221(96)00030-2Search in Google Scholar

[42] C. Meyer, A. Rademacher and W. Wollner, Adaptive optimal control of the obstacle problem, SIAM J. Sci. Comput. 37 (2015), no. 2, A918–A945. 10.1137/140975863Search in Google Scholar

[43] C. Miehe, F. Welschinger and M. Hofacker, Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations, Internat. J. Numer. Methods Engrg. 83 (2010), no. 10, 1273–1311. 10.1002/nme.2861Search in Google Scholar

[44] A. Mikelić, M. F. Wheeler and T. Wick, Phase-field modeling through iterative splitting of hydraulic fractures in a poroelastic medium, GEM Int. J. Geomath. 10 (2019), no. 1, Paper No. 2. 10.1007/s13137-019-0113-ySearch in Google Scholar

[45] I. Neitzel, T. Wick and W. Wollner, An optimal control problem governed by a regularized phase-field fracture propagation model. Part II: The regularization limit, SIAM J. Control Optim. 57 (2019), no. 3, 1672–1690. 10.1137/18M122385XSearch in Google Scholar

[46] J. Nocedal and S. J. Wright, Numerical Optimization, Springer Ser. Oper. Res. Financ. Eng., Springer, New York, 2006. Search in Google Scholar

[47] R. Rannacher and J. Vihharev, Adaptive finite element analysis of nonlinear problems: Balancing of discretization and iteration errors, J. Numer. Math. 21 (2013), no. 1, 23–61. 10.1515/jnum-2013-0002Search in Google Scholar

[48] T. Richter and T. Wick, Variational localizations of the dual weighted residual estimator, J. Comput. Appl. Math. 279 (2015), 192–208. 10.1016/j.cam.2014.11.008Search in Google Scholar

[49] F. Suttmeier, Numerical Solution of Variational Inequalities by Adaptive Finite Elements, Vieweg+Teubner, Wiesbaden, 2008. Search in Google Scholar

[50] A. Veeser, Efficient and reliable a posteriori error estimators for elliptic obstacle problems, SIAM J. Numer. Anal. 39 (2001), no. 1, 146–167. 10.1137/S0036142900370812Search in Google Scholar

[51] T. Wick, Goal functional evaluations for phase-field fracture using PU-based DWR mesh adaptivity, Comput. Mech. 57 (2016), no. 6, 1017–1035. 10.1007/s00466-016-1275-1Search in Google Scholar

[52] T. Wick, Multiphysics Phase-Field Fracture: Modeling, Adaptive Discretizations, and Solvers, De Gruyter, Berlin, 2020. 10.1515/9783110497397Search in Google Scholar

Received: 2020-03-21
Revised: 2021-05-08
Accepted: 2021-05-11
Published Online: 2021-06-05
Published in Print: 2021-07-01

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